Find the slope and the $y$-intercept of the line.
$-8 x+4 y=-4$
Write your answers in simplest form.
slope: $\square$
$y$-intercept: $\square$
Answer (1)
Rewrite the given equation in slope-intercept form: y = m x + b .
Isolate the y term: 4 y = 8 x − 4 .
Divide both sides by 4: y = 2 x − 1 .
Identify the slope as 2 and the y -intercept as -1: 2 , − 1
Explanation
Understanding the Problem We are given the equation of a line: − 8 x + 4 y = − 4 . Our goal is to find the slope and the y -intercept of this line. To do this, we will rewrite the equation in slope-intercept form, which is y = m x + b , where m is the slope and b is the y -intercept.
Isolating the y term First, we want to isolate the y term. We can add 8 x to both sides of the equation: − 8 x + 4 y + 8 x = − 4 + 8 x 4 y = 8 x − 4
Solving for y Next, we divide both sides of the equation by 4 to solve for y :
4 4 y = 4 8 x − 4 y = 4 8 x − 4 4 y = 2 x − 1
Identifying Slope and y-intercept Now that the equation is in slope-intercept form, y = 2 x − 1 , we can easily identify the slope and the y -intercept. The slope, m , is the coefficient of x , which is 2. The y -intercept, b , is the constant term, which is -1.
Final Answer Therefore, the slope of the line is 2 and the y -intercept is -1.
Slope: 2 y -intercept: − 1
Examples
Understanding the slope and y-intercept of a line is crucial in many real-world applications. For example, if you are tracking the cost of a taxi ride, the slope represents the rate per mile, and the y-intercept represents the initial fee. Similarly, in physics, the slope of a velocity-time graph represents acceleration, and the y-intercept represents the initial velocity. By understanding these concepts, you can easily model and analyze linear relationships in various fields.
